Absolute Trust: Algorithm for Aggregation of Trust in Peer-to- Peer Networks

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by  Sateesh Kumar Awasthi

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Absolute Trust: Algorithm for Aggregation of
Trust in Peer-to- Peer Networks
Sateesh Kumar Awasthi, Yatindra Nath Singh, Senior Member, IEEE,
Abstract—Tomitigatetheattacksbymaliciouspeersandtomotivatethepeerstosharetheresourcesinpeer-to-peernetworks,
severalreputationsystemshavebeenproposedinthepast.Inmostofthem,thepeersevaluateotherpeersbasedontheirpast
interactionsandthenaggregatethisinformationinthewholenetwork.Howeversuchanaggregationprocessrequiresapproximations
inordertoconvergeatsomeglobalconsensus.Itmaynotbethetruereflectionofpastbehaviorofthepeers.Moreoversuchtypeof
aggregationgivesonlytherelativerankingofpeerswithoutanyabsoluteevaluationoftheirpast.Thisismoresignificantwhenallthe
6 peersrespondingtoaquery,aremalicious.Insuchasituation,wecanonlyknowthatwhoisbetteramongthemwithoutknowingtheir
1 rankinthewholenetwork.Inthispaper,weareproposinganewalgorithmwhichaccountsforthepastbehaviorofthepeersandwill
0 estimatetheabsolutevalueofthetrustofpeers.Consequently,wecansuitablyidentifythemasagoodpeersormaliciouspeers.Our
2 algorithmconvergesatsomeglobalconsensusmuchfasterbychoosingsuitableparameters.Becauseofitsabsolutenatureitwill
equallyloadallthepeersinnetwork.Itwillalsoreducetheinauthenticdownloadinthenetworkwhichwasnotpossibleinexisting
n
algorithms.
a
J
IndexTerms—Peer-to-peerNetwork,Trust,GlobalTrust,LocalTrust,DHT,Non-negativematrix,Eigenvector.
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1 INTRODUCTION
.
s
c FOR exchanging and sharing the information, peer-to In such a network, consensus is estimated by the manager
[
peer networks are better because of their inherent ad- peer. In an unstructured network, each peer evaluate the

1 vantage of scalability and robustness, as compare to tra- globaltrustvalueofpeersbycollectingthelocaltrustfrom
v ditional client server model. Every peer in p2p network differentpeersthroughadistributedaggregationalgorithm,
9 can initiate the communication and each peer can act both theaggregationcanbedoneeitherbygossipingprotocolor
1 like client as well as server, and has equal responsibility. bytakingfeedbackonlyfromfewsignificantpeers;however
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But due to lack of functionality of central control, some taking feedback only from few of the peers does not make
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peerscaneasilysabotagethenetworkbyputtinginauthentic the global trust, global in true sense. In both structured
0
. contents in the network. Such peers are called malicious and unstructured network, if consensus is taken across the
1
peers. Furthermore, rational behavior of peers encourage whole network, then local trust needs to be normalized in
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them only to draw the resources from network without some way, which results in approximation of global trust.
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1 sharinganything.Thesetypesofpeerarecalledfreeriders. Some times it may not be the true reflection of the past
: Insuchasituation,p2pnetworkfunctionslikeapoorclient behavior of peers. This type of aggregation gives only the
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serversystemwhereonlyfewpeersactasserverwithmuch ranking of peers. This ranking system is similar to the
i
X less upload bandwidth and storage capacity. The success randomsurferModel[17]whichisbasedonthepopularity
r of peer to peer networks largely depends on the policy by of page on the web. But there is a difference between
a which these two issues can be handled. Many researchers popularity and trustworthiness; a peer can be popular in
have proposedto implement a reputationsystem basedon anetworkbydoingtransactionwithlargenumberofpeers,
the past behavior of peers in the network. Past behavior is butmaynotbeprovidinggoodqualityofservice.However,
modelledastrust.Itisqualitativeanddifficulttomeasurein for a trustworthy peer, quality of service provided in each
quantity.Hence,suitablemetricandalgorithmsarerequired transactionwillbegood.
to measure as well as propagate it to all the nodes, so that Let us understand it by following example. Let there
behaviorofpeerscanbemodelled. be five peers in a network - A, B, C, D and E. After some
Inmost of the existing reputationsystems,all the peers interactionstheygivesomelocaltrustvaluetoeachotheras
evaluatetheotherpeers,basedonthepastinteractionsand shown in table I. After aggregating these local trust values
assign them some trust value, also called local trust value. [3][6],theyarerankedasB,E,C,D,A;Bismosttrustworthy
These local trust values are basic information, which are andAisleasttrustworthy.Ifitisaggregatedasin[10],then
aggregatedinwhole networktoform theglobalreputation theyarerankedasE,B,D,C,A;EismosttrustworthyandA
of the peer. This aggregation process is different for struc- isleasttrustworthy.ButwecanseeclearlyfromTable1that
turedandunstructuredp2pnetwork.Instructurednetwork, Aismakingtwotransactionsbutbothtransactionsaregood
responsibility to manage global reputation through aggre- ascomparedtoanytransactionmadebyB,CandD.Sowe
gation is distributed among all the peers. It is also called cannot conclude thatA islesstrustworthyascomparedto
globaltrustvalueofpeer.WiththehelpofDHTalgorithms, B,CandD.
suchasChord[12],CAN[13],Pastry[14],Tapestry[15],the Furthermorerankingofpeersdoesnotgiveanyabsolute
peer managing global trust of a peer can be easily located. characterization of them. There are many situation where
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Table 1: Local trust of peers A, B, C, D and E, zero means
section 3, we will define the basic trust model and its ag-
thereisnointeractionbetweenpeerstillnow
gregatingalgorithm.Section4willbecoveringtheexistence
and uniqueness of proposed global trust. In section 5, the
A B C D E
algorithmisanalyzed.Section6presentssimulationresults,
A 0 0.6 0.6 0 0 and finally in section 7, conclusion and future work is
presented.
B 0.3 0 0.3 0.4 0.4
C 0.4 0.4 0 0.2 0.2
D 0.5 0.1 0.1 0 0.5
2 RELATED WORK
E 0.7 0.7 0.8 0 0
Reputationsystem is used to establishthe trust among the
buyersine-commercee.g.Amazon,Flipkart,Snapdeal,eBay
we need absolute characterization of peers. Consider the
[1]. Inallsuchsystems,there is somecentralauthorityand
examplewhenapeersendarequestforaparticularfileand
it is keeping the record of past experiences of buyers. This
alltherespondingpeersaremalicious,thenrankingsystem
experience is used by new buyers for their shopping. Ag-
can only tells us that who is better among them. We will
gregating the feedback in the presence of central authority
neverknowwhethertheyaremaliciouspeersorgoodpeers.
is simple task, but p2p system is distributed in nature so
Itcanresultininauthenticdownloadinthenetwork.Besides
maintainingandaggregatingthetrustisnottrivial.
all these facts, relative ranking of peers overload the most
Aberer and Despotovic [2] proposed a trust model in
reputable peers even if we use the probabilistic approach
which only complaints are reported if any, otherwise peers
to select the source peer for download, i.e.,the probability
are assumed to be trustworthy. Eigentrust Algorithm [3] is
of choosing a download source peer is proportional to its
based on Random Surfer Model [17]. Pre-trusted peers are
global trust. Also, the process of normalization method is
required to handle the malicious peers in it. In PeerTrust
message consuming task. For example, if trust assigning
[4] five different factors are defined for evaluationof trust-
peerupdatesthelocaltrustofanyoneofitsinteractingpeer,
worthiness of the peers. Both Eigentrust and PeerTrust are
thenitneedstoupdatethevaluesofalltheotherinteracting
based on the concept of weighted average. Fuzzy Trust
peers. It will require more messages to communicate the
model [5] proposed by Song et al. It is also based on
updatetothetrustholderpeers.
the concept of weighted average, where weight factor is
Keepinginviewalltheabovepoints,areputationsystem
determined by three variables– the peer’s reputation, the
in p2p network must have the following design considera-
transaction date and the transaction amount. The message
tions.
complexityinFuzzyTrust[5]islesserthanintheEigentrust
• Reputationshouldbetruereflectionofpastbehavior. [3]. PowerTrust [6] is based on assumption of the power
• Reputation must be aggregated in the whole net- law network. In it local trust is aggregated similarly to the
work. Eigentrust[3]exceptpre-trustedpeersarereplacedbymost
• The system should be robust to the malicious peers reputable peers in the network. These reputable peers are
withasmanyattackersmodelaspossible. searchedandelecteddynamicallyinthenetwork.Allabove
• LoadBalance: Systemshouldnot overloadonly few trustmodels[2][3][4][5][6]areforstructurednetworkand
peersinnetwork. DHTisusedforefficientlocationoftrustholderpeers.
• Adaptivetopeerdynamics
In unstructured network, global trust is calculated by
• FastConvergenceSpeed
floating the query for local trust in the network. The peer,
• Loweroverhead/messagecomplexity
who wants to calculate the global trust, waits for the feed-
• NoCentralauthority
back upto some time. Then the calculation of global trust
In this paper, we propose a metric and an aggregation is performed with these limited number of feedback given
algorithm which truly capture the past behavior of the by some of the peers. Gossip Trust [7] used same metric
peers.Theproposedaggregationalgorithmdoesnotrequire as in [3] and local trust valuesare gossipedin the network
anykindof normalizationhence it automaticallymeetsthe similarly to randomized gossip algorithm in [17]. In Scal-
above design considerations. It is purelydecentralized and ableFeedbackAggregation(SFA)[9],thetrustworthinessis
doesnotrequireanykindofcentralauthorityorpre-trusted calculatedby weighted average of local trust and feedback
peers or power nodes. The Absolute Trust is based on the takenbyfewofthepeers.Antoninoetal.proposedaflow-
conceptofweightedaveragingandscalingoflocaltrust.Itis basedreputation[10] whichis modified versionof [3]. It is
calculatedrecursivelyinthewholenetworktillitconverges. only for centralize systems. Wang and Vassileva proposed
We will show that it will converge at some unique global a Bayesian Trust Model [24] in which, different aspect of
value and can be calculated distributively in the whole peerbehavioraremodelledindifferentsituations.Damiani
networkbyallthepeers.Oursimulationresultsshowthatit et al proposed a system [25] for managing and sharing
gives betterauthenticdownloading performance and more the servent’s reputation in which peers poll other peers
uniform load distributions among good peers with lesser by broadcasting a request for opinion. In another similar
messagecomplexity. approach [26] Damiani et al. considered the reputation of
Rest of the paper is organized as follows. Section 2 both peers and resources, but credibility of voter was not
represents the past work done on reputation systems. In consideredinboththeapproaches.
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3 PROPOSED TRUST MODEL
Inthismodelofpeertopeernetwork,thepeersareassumed
to exchange only the files as the resource . With suitable
modification, the same model can also be used for other
kind of resources. We will define the basic trust metric,
namely local trust, which is the raw data used for the
calculation of global trust, which the trust, system as a
whole keeps on an individual peer. Later we will give an
algorithmfortheaggregationofthelocaltrustsinthewhole
networktogenerateglobaltrustvalue.
(a)One-to-Many (b)Many-to-One
3.1 LocalTrust
Typically peer’s satisfaction after a transaction can be clas-
sifiedassatisfied,neutralorunsatisfied.Wecanalsodefine
manyotherlevels, butfor simplicityonlythree levels have
beenassumed.Letpeeridownloadsomefilesfrom peerj,
thenpeericanassignalocaltrustvaluetopeerj as
n w +n w +n w
g g n n b b
T = .
ij
n
t
(c)One-to-One
where,
Figure1:Differentwaysofevaluation
n =Numberofsatisfactroyfiles,
g
n =Numberofaverageorneutralfiles,
n
n =Numberofunsatisfactroyfiles,
b
n =TotalNumberofdownloadedfiles, aggregationprocesswillremainsame.Incaseoffreeriding,
t
w =Weigthtfactorforsatisfactroyfiles, one can define the metric for local trust in many different
g
w =Weightfactorforaverageorneutralfiles, ways. In the next section, above issues are not discussed
n
w =Weightfactorforunsatisfactroyfiles, and we focus only on the information aggregation in the
b
network.
Let us assume that the variation of weight factor varies
linearlyfromunsatisfactoryfiletosatisfactoryfile,then
w +w 3.2 AbsoluteTrust:AlgorithmforAggregation
g b
w = .
n
2
Inanyevaluationprocess,therearetwoparties,onewhois
Onsimplification, evaluating;wewillcallittheevaluator,andtheonewhois
n w +(n −n −n )(wg+wb) +n w being evaluated; we will call it the evaluatee. Reliabilityof
T = g g t g b 2 b b evaluationdependsonwhoisevaluating,anditvariesfrom
ij
n
t person to person. It is said to be more reliable if it is done
(cid:2) (cid:3)
byacompetentevaluator.
1
T = [(n −n +n )w +(n −n +n )w ]
ij 2n g b t g b g t b There are three different scenario in the evaluation as
t
shown in figure 1. One-to-many: one person is evaluating
1
T = [(x−y+1)w +(y−x+1)w ] (1) many persons; many-to-one: many persons are evaluating
ij g b
2 one person; and one-to-one: one person is evaluating an-
where, other person. In one-to-many scenario, since evaluation is
x=Fractionofsatisfactoryfiles,and done byonlyone person, the evaluationcanbe considered
y =Fractionofunsatisfactoryfiles. to be uniform. Furtherthe evaluatedmetriccan be linearly
scaled up or down. In many-to-one evaluation since one
Thismetricwillensurethatlocaltrustvaluewillremain person is evaluated by many persons so there are chances
between w and w . For example if peer i download 100 of contradictions. At the same time, the opinion of any
g b
filesfrompeerAandB,andAprovide20satisfactoryfiles, evaluatorcannotbeignored. Thus,thebestwaytoresolve
40 unsatisfactory files and rest average files while peer B the contradiction is to take the weighted average of all the
provide 30 satisfactoryfiles, 60 unsatisfactoryfiles and rest evaluators’opinions,whileassigningmoreweighttoamore
averagefiles;assumingthatweight factor ofgood file is 10 competent evaluator. In one-to-one evaluation, there is no
andthatisforbadfileis1,thenT =4.6andT =4.15 directcomparisonoftwoevaluationsbecausetheevaluator
iA iB
Many author argue that there are many other factors andevaluateebothare different. Inorder tocompare these
which can influence the local trust value like amount of evaluationsitisessentialtomakethemuniformwithrespect
transactions, date of transactions, number of transactions to evaluator. Again based on the concept that competent
etc. [4] [5] [9] [11]. We agree with their arguments which evaluator’s evaluation will be more accurate, we can bias
can also be considered in our case. But in all the cases, the these evaluationby a weight factor which must be propor-
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tional in some sense to the competence of evaluator. This
10
biascanbegivenby
w =1
e
Eval uniform out=[(Eval value in)p.(we)q]p+1q (2) we =2
8 w =3
where, Eval value in is evaluationdone by an individual ut we =4
o e
evaluator, w is weight factor assigned to this evaluator,
e m we =5
Eval uniform out is output uniform evaluation and p , r 6 w =6
o e
qarIefspu=itaqb,lythcehnosEenvaclonusntaifnotsr.m out is geometric meanof nif we =7
u w =8
w andEval value in.Ifwetakeq =α.pthen e
e al 4 we =9
v
Eval uniform out=[(Eval value in).(we)α]1+1α E we =10
The impact of w and α can be seen in figure 2(a), (b) and
e 2
(c).Inthesefigures,wecanseethatEval uniform outin-
creasesfasterwithincreasingEval uniform inforhigher
2 4 6 8 10
valuesofw andforlowervaluesofα. Thetransformation
e
suppresses the reputation reported by less reputed peers, Eval value in
because the suppression is higher for lesser weight factor (a)α=1/2
w . Also Eval uniform out is monotonically increasing
e
withEval uniform in. 10
Now let there be N peers in the network, and they are we =1
interactingwitheachother.Ifanypeeritakestheserviceof we =2
anypeerjthenicanevaluatej′strustaccordingtoequation 8 we =3
t
1.Eachicanevaluateallsuchj independentlyandthereis u w =4
o e
no need of any modification in them, because it is one-to- m we =5
many evaluation. We are aggregating the values of these r 6 w =6
o e
one-to-many evaluations (local trust) resulting in estimate nif we =7
ofabsoluteglobaltrust. u w =8
e
al 4 we =9
Each peer is also providing services to many other v
E we =10
peers, and is being evaluated by them. This is many-to-
one evaluation. The aggregated trust values after this step
2
willbeweightedaverageofallthelocaltrustestimates.The
weight factor can be chosen in many different ways, but
global trust of an individual peer will be best choice to be 2 4 6 8 10
used as a weight. Many authors argue that a good service Eval value in
provider may not be a good feedback provider [4] [9] [11]. (b)α=1/3
But we argue that until peers are not in the competition, a
good service provider will be most likely a good feedback 10
provider. So we have taken global trust of peers as the w =1
e
weight factor for the purpose of aggregation of local trust. w =2
e
Henceglobaltrust,ti ofanypeeriisgivenby t 8 we =3
u w =4
T t o e
ti = j∈Si ji j ∀i. (3) m we =5
P j∈Sitj or 6 we =6
Here,Si isasetofpeePrsgettingservicesfrom peeri,Tji is nif we =7
localtrustofpeerievaluatedbypeerj,tj isglobaltrustof u we =8
peerj.Equation3canberearrangedas al 4 we =9
v
E we =10
T t
t = j∈Si ji j
i e .C.t
P i 2
= (e Ct)−1T t
i ji j
jX∈Si 2 4 6 8 10
Eval value in
These set of N equations can be written in the form of
matrixas (c)α=1/4
t=[diag(e Ct,e Ct,.....e Ct)]−1Ttt Figure 2: Transformation curve, taking evaluated value of
1 2 N
trust as a input and uniform evaluated trust value as a
where, t is global reputation vector, T is trust matrix, its outputshownfordifferentvaluesofα
T element is local trust value of peer j assigned by peer
ij
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i. The element T is zero if there is no interaction among
ij
peeriandpeerj,eiisrowvectorwithithentryas1andall t=(D.Tt.t)1+1α.
others are zero, C is incidence matrix corresponding to Tt
im.ea.tirfixTjTi.>0,thenCij =1,elseCij =0.Tt istransposeof D(ei(.eiCsi..Cda.ita)g(d1(t+i)aα.tg))oαna,dl iamga(ttr)ixi,sNwXithN ditisagiotnhalemleamtreinx,twditihiatss
It is clear from equation 3 that value of ti will remain (cid:2)iith elementast(cid:3)i andα=q/p.Restallhavesamemeaning
betweenminimumandmaximumvalueoflocaltrustgiven as mentioned above. Power of the vector is defined as the
bypeersbelongingtoS. powerofitsindividualelement.
Nowinthewholenetworkeverypeerisevaluatedbya In this set, there are N unknowns and N non-linear
different set. If the sets can be represented equivalently as equations, hence we can not state any thing directly about
a singlepeer, thenit is sameasone-to-one evaluation.This the solution of these equations. However we will show in
evaluation can be made uniform using equation 2. To give next section that there exist a unique positive global trust
the equivalent globaltrust of the set, consider a set S of m vector,correspondingtothesesetofequations.Wecanfind
peers with global trust valuest ,t ......t . The globaltrust the solution iteratively. In each iteration, we are taking a
1 2 m
ofthesetmustbedominatedbythemoretrustworthypeers wider view of global trust of any peer i in the network.
because we are giving more weight to their opinion. With Insubsection5.1,speedofconvergencehasbeendiscussed.
the notion of weighted average, intuitively, we can define We can see directly from figure 3 that the convergence is
theglobaltrustofthesetas more rapid in the initial iterations. This shows that, in the
calculationofglobaltrustwearegivingmoreweightageto
t2
j∈S j the one hop neighbors andthe weightagedecreasesashop
t = . (4)
s
t countsareincreasing.
Pj∈S j
ThisequationissimilartoeqPuation3.Here,weareensuring
that global trust of a set will be dominated by the peers 4 EXISTENCE AND UNIQUENESS OF GLOBAL
having higher global trust value. It will always be in be- TRUST
tweentheminimumandmaximumvaluesofglobaltrustof Weareproposingfollowing lemmasandtheoremstoshow
the members of set S. The global trust, ti, of a peer i, can the existence and uniqueness of global trust vector. Few
be biased by the global trust, tsi, of trust assigning set Si definitionswhichwillbeusedinthissection.
according to equation 2, then the modified global trust of
peericanbewrittenas
Definition 1. A vector v or matrix M is said to be
ti =[tip.tsiq](p+1q) ; ppoossiittiivvee//nnoonnnneeggaattiivveeainfdirtseale.ach element vi or Mij is
T t p t2 q (p+1q) Definition2. Avectorv′/matrixM′ issaidtobelessthan
ti =" Pj∈jS∈iSijtij j! . Pjj∈∈SSiitjj! # (5) v′′/M′′ ifitseachelementvi′/Mi′j islessthanvi′′/Mij′′.
Equation 5 wiPll be the true rePflection of past behavior Lemma 1. LetzbeapositivevectorinRN,suchthatz=f(t),
of peer i in the whole system. This equation will give us α
withitsith elementas ti(eiCt) .t .Then∃atleastone
theabsoluteinterpretationofglobaltrustvalueofanypeer. (eiC.diag(t).t) i
We have made it uniform by using a biasing factor t . We (cid:20)′ ′′ (cid:21) ′′ ′
s pairofpositivevectorst andt suchthat,ift >t ,then
cannowdirectlycomparetheglobaltrustvaluesofanytwo
′′ ′
peers.Equation5canberearrangedas f(t )>f(t )
q/p 1 whereαisanarbitraryrationalnumber.
T t t2 (1+q/p)
ti =" Pj∈jS∈iSijtij j!. Pjj∈∈SSiitjj! # PWrohoefr.eLeetisusacvoencstiodrerwtiwthoavlelcetloermsetn′t=s aas.e’1a’,nadatn′′d=bba.ree.
P P 1 scalarsuchthatb>a>0.Thenith elementofvectorf(t′)
( t2)q/p (1+q/p)
=" ( Pj∈jS∈iStij)j(1+q/p)!. jX∈SiTjitj!# fi(t′)= (e Ct.′id(eiaiCg(tt′)′).t′) α.t′i
P 1 (cid:20) i (cid:21)
(e .C.diag(t).t)α (1+α) a(m.a) α
= i . T t f (t′)= .a
" (ei.C.t)(1+α) ! jX∈Si ji j!# i (cid:20)(m.a2)(cid:21)
′
f (t)=a
1 i
= (ei.C.diag(t).t)α . T . t (1+α) heremisnumberof′1′ inith rowofincidencematrixC.
"jX∈Si (ei.C.t)(1+α) ! ji! j!# Similarly ′′
f (t )=b
There are N nodes in the network, i = 1,2,.....N, so i
′ ′′
thesesetofN equationscanbewrittenintheformofmatrix hence ∃ a pair of positive vectors t and t satisfying the
asfollows condition.
6
where a ,a and a are some scalers, adding 6, 7 with all
1 2 3
combinationsof8willresult
Lemma 2. Let A and B be NXN non negative, irreducible
matriceswith spectralradius’1’,andcorrespondingeigenvector lim (A−B)k−1x≈bv (9)
v. Then for any vector x; having at least one component along k→∞
vectorv. herebisalinearcombinationofa1,a2 andalla3.Nowpre-
multiplyingequation9by(A−B),
lim (M .M .M ........M )x=c.v
k→∞ 1 2 3 k (A−B)(A−B)k−1x=(A−B)bv=(v−v)b=0
HereMicanbeAorBforallifrom1tokandcisanyscalar.A
Hence
andBaresuchthat(M1.M2.M3........Mk)isalsoirreducible. lim (A−B)kx=0
k→∞
Proof. Let the eigen vectors of matrix A and B are
v,v2,v3,......vN and v,u2,u3,.....vN. Then any vector x;
having at least one component along vector v, can be Theorem 2. Let A,A1,A2......Am be NXN non negative,
expressedas irreduciblematrices,withspectralradius1,λ ,λ ......λ respec-
1 2 m
tively.Letthecorrespondingeigenvectorforalltheabovematrices
x=a v+a v +.......a v
1 2 2 N N bev.Thenforanyvectorx.
and lim (A +A +.....A −A)kx=0
x=b1v+b2u2+......bNuN k→∞ 1 2 m
whenthis vectorwillpassthroughmatrix AandB thenit if|λ1+λ2+......λm−1|<1
willbe Proof. Let
M=(A +A +.......+A )
Ax=a v+a λ v +.....+a λ v 1 2 m
1 2 2 2 N N N
then
and
M.v=(A +A +.......+A ).v
Bx=b v+b γ u +......+n γ u 1 2 m
1 2 2 2 N N N
=(λ +λ +......+λ ).v=λ.v
1 2 m
where λ ,λ ......λ and γ ,γ ......γ are eigen values of
2 3 N 2 3 N
matrix A and B respectively. If it will again pass through hencevisalsoaneigenvectorofmatrixMandcorrespond-
any of A and B, then vector v will remain as it is ing eigen value is λ. Matrix M is the sum of non negative,
and magnitude of all other vectors will decrease because irreduciblematricesA1,A2......Am thereforeMisalsonon
1 > |λ | > |λ |.... > |λ | and 1 > |γ | > |γ |...... > |γ | negative and irreducible. So we can conclude that spectral
2 3 N 2 3 N
(see[16]).Thus radiusofmatrixMisλ.
FurtherMcanbewrittenas
BAx=a v+δav+L.O.M.O.u ,u ,....u
1 2 3 N M (λ−1)M
M= +
and λ λ
(cid:20) (cid:21)
ABx=b v+δbv+L.O.M.Ov ,v ,......v =[B+N]
2 2 3 N
L.O.M.O. means ”lower order magnitude of”. Repeating here B is M/λ and N is (λ − 1)M/λ. Matrix B and N
thisoperationkth timesinanyorderwewillget are scalar multiple of non negative irreducible matrix M
therefore matrix B and N also follow the properties of
lim (M .M .M ........M )x=c.v
1 2 3 k non negative irreducible matrices. Hence spectral radius
k→∞
of matrix B and N is ’1’ and |λ − 1| respectively and
whereMi canbeAorBforallifrom1tok correspondingeigenvectorisv.
If|λ−1|<1then
Theorem 1. Let A and B be NXN non negative, irreducible lim Nkx=0, (10)
matriceswith spectralradius’1’,andcorrespondingeigenvector k→∞
v.ThenforanyvectorxinRN andfromTheorem1
lim (A−B)kx=0 lim (B−A)kx=0. (11)
k→∞
k→∞
In fact when vector x is passed through any of N or
Proof. InLemma2letMi =Aforalli,then (B−A)itsmagnitudedecreases,andatk →∞,itbecome
lim Ak−1x≈a v, (6) zero.Soingeneralwecanwrite
1
k→∞
andifMi =Bforalli,then kl→im∞(M1.M2.M3........Mk)x=0 (12)
lim Bk−1x≈a2v (7) where Mi can be any of N or (B−A). Adding all the
k→∞ combinationsofequation12withequation10andequation
ifMi istakenrandomlyAorB,then 11,wewillget
lim (A.B......B.A...(k−1)times)x≈a v (8) lim (N+(B−A))kx=0
3
k→∞ k→∞
7
or where tk is the value of vector t in kth iteration and φ is the
lim (M−A)kx=0 iterative function from RN → RN. The error in vector t will
k→∞ convergebythefactor 1+α ineveryiteration.
α
hence
lim (A1+A2+.....Am−A)kx=0 Proof. Letusrearrangetheiterativefunctionφ(tk−1)
k→∞
if|λ−1|<1or|λ +λ +......λ −1|<1. tk =φ(tk−1)
1 2 m
=[D(tk−1).Tt.tk−1)]1+1α (13)
Theorem 3. let T be NXN non negative, irreducible matrix =[diag(d1,d2....dN)Tt.tk−1]1+1α
then,∃apositivevectortsuchthat
where
(t)1+α =(D.Tt.t)
e C.diag(tk−1).tk−1 α
i
DeiCis.diadgia(tg)o.tnaαl .matrix, with its ith element di as di = (cid:0) eiCtk−1 (1+α) (cid:1) !
(1+α)
(cid:20)(cid:0) eiCt (cid:1) (cid:21) Thenith elementoftk w(cid:0)illbe (cid:1)
Proo(cid:0)f. Re(cid:1)lation(t)1+α =(D.Tt.t)canbewrittenas
Ttt=D−1(t)1+α =y e Tt.tk−1 e C.diag(tk−1).tk−1 α 1+1α
tk = i i
awshereyi =(cid:20)(ei(Cei.Cdita)g(1(+t)α.)t)α(cid:21)t1i+α.Further,yicanbewritten i "(cid:0)eiTt.tk−1(cid:1)(cid:0)(cid:0)e1+i1Cαteki−C1.(cid:1)d1+iaαg(tk−1).tk(cid:1)−1# 1+αα (14)
(e Ct)α =
yi =(eiCt) (e C.diiag(t).t)α t1i+α "(cid:0) (cid:1) (cid:0)eiCtk−1 (cid:1) #
(cid:20) i (cid:21)
t (e Ct) α Lettki andtik−1are,far(cid:0)fromactu(cid:1)alsolutiontibyδtki and
=(eiCt) e Ci.diiag(t).t ti δtik−1 respectively,then
(cid:20) i (cid:21)
=(eiCt).fi(t) t +δtk = eiTt.(t+δtk−1) 1+1α .
=ei(fi(t)C).t i i "(cid:0) eiC(t+δtk−1(cid:1)) #
hencevectorycanbewrittenas [(cid:0)eiC.diag(t+δt(cid:1)k−1).(t+δtk−1) 1+αα]
y=F(t).t (cid:0) (cid:1)
wmhaterriexmCaatrnidxFth(etr)ehfoarsenaotnsazmeroepeloesmiteionntsaastTsatm,itespijoseilteiomneanst = eiTt.t 1+1α eiC.diag(t).t 1+αα . (1+ ei.eTit.T.δtt.kt−1)1+1α .
Fij(t)isfi(t),hence "(cid:0) (cid:1) (cid:0)eiCt (cid:1) #" (1+ ei.eCi..Cδt.kt−1) #
Tt.t=F(t).t 2ei.C.diag(t)(cid:0).δtk−(cid:1)1 ei.C.diag(δtk−1).δtk−1 1+αα
1+ +
Now,∃apositivevectort′,suchthatf (t′)≤min(T >0). "(cid:18) ei.C.diag(t).t ei.C.diag(t).t (cid:19) #
i ij
′
Forsucht , Since δtk−1 << t hence we can neglect the higher order
Tt.t′ >F(t′).t′. termsofδtk−1.
ForAsulscoh∃t′a′,positivevTetc.tto′r′ <t′′,Fs(utc′′h).tth′′atfi(t′′)≥max(Tij), ≈"(cid:0)eiTt.t(cid:1)1+1α(cid:0)eeiiCC.tdiag(t).t(cid:1)1+αα#."(1(+1+ei.eTeiit..TeC.δitt..Cδ.ktt−.kt1−)11)+1α#.
function f is continuous and from Lemma 1 there exist a (cid:0) (cid:1) 2ei.C.diag(t).δtk−1 1+αα
fp(att′h).frHoemncfe(,t∃′)atopof(stit′′iv)esuvcehctothrattbifettw′′e>entt′,′tahnedntf′(′t,′s′u)c>h "(cid:18)1+ ei.C.diag(t).t (cid:19) #
that Usingequation14
Tt.t=F(t).t
hence∃apositiveve(ctt)o1r+tα,=suc(hDt.hTatt.t) =ti."(1(+1+ei.eTeiit..TC.δtt..δktt−k1−)11)+1α#.
ei.C.t
2ei.C.diag(t).δtk−1 1+αα
1+
Theorem4. Vectortintheorem3isuniqueandcanbecalculated "(cid:18) ei.C.diag(t).t (cid:19) #
byiterativefunction
Using binomial expansion and neglecting higher order
tk =φ(tk−1)=[D(tk−1).Tt.tk−1)]1+1α, termsofδtk−1
8
(1+ ei.Tt.δtk−1 ) = [X + Y − Z].δtk−1
≈t . (1+α)ei.Tt.t . i i i
i " (1+ ei.eCi..Cδt.kt−1) # WhereXi,Yi andZi areith rowofNXN matrixX,Yand
2αe .C.diag(t).δtk−1 Zrespectively,itisclearfromtheabovethat
i
1+
"(cid:18) (1+α)ei.C.diag(t).t(cid:19)# Xt= 1 .t,
1+α
2α
δtki =ti. Yt= 1+α.t
"(cid:0)11++(e1ei+i..TeCαit)...eCδδitt..ktkT−−t11.t(cid:1)(cid:18)1+ 2(1αe+i.αC).edii.Cag.d(ti)a.gδt(kt)−.1t(cid:19)#−ti and Zt=1.t,
where,
(cid:0) (cid:1)
δtki =ti. X,Y,Z,t>0
1+ (e1i+.Tαt).eδit.kT−t1.t 1+ 2αei.C.diag(t).δtk−1 −1 Matrices X,Y,Z have non zero elements at same position
"(cid:0)(1+ ei.eCi..Cδt.kt−1)(cid:1)(cid:18) (1+α)ei.C.diag(t).t(cid:19) # aspsemctartarlixraTditu,shoenfcXe,aYlltahnedseZawreilallbsoeirr1ed,uc2iαblea.nTdh1e.refore
1+α 1+α
Now
δtk =(X+Y−Z)δtk−1
=t .
i
e .Tt.δtk−1 2αe .C.diag(t).δtk−1 Ifinitialerrorintisδt0 then
i i
1+ 1+
" (1+α)ei.Tt.t! (1+α)ei.C.diag(t).t! limk→∞δtk =limk→∞(X+Y−Z)kδt0
e .C.δtk−1 e .C.δtk−1
i i
− 1+ 1+ DirectlyfromTheorem2
ei.C.t !#, ei.C.t !
limk→∞(X+Y−Z)kδt0 =0
Approximatingthedenominatorterm 1+ei.C.δtk−1 ≈1
ei.C.t
(cid:18) (cid:19)
⇒limk→∞δtk =0
≈ti. if
1 2α
e .Tt.δtk−1 2αe .C.diag(t).δtk−1 | + −1|<1
i + i 1+α 1+α
" (1+α)ei.Tt.t! (1+α)ei.C.diag(t).t! α
⇒ <1
e .Tt.δtk−1 2αe .C.diag(t).δtk−1 1+α
i i
+ .
(1+α)ei.Tt.t! (1+α)ei.C.diag(t).t! Whichistrueforanyα>0
− ei.C.δtk−1 In every step error will decrease by a factor of 1+αα .
ei.C.t !# Vector t will converge fast if value of α is small. Hence
speedofconvergencewillbe
Againneglectinghigherordertermsofδtk−1
1+α
=
α
≈t .
i
ei.Tt.δtk−1 2αei.C.diag(t).δtk−1 Intheorem4ifα=q/pthenith elementofvectortwill
+
" (1+α)ei.Tt.t! (1+α)e−i.C .edii.aeCgi..C(δtt).kt.−t1!!# be ti = (cid:0)ei(Cei.dCita)g(1(+t)pq.t)(cid:1)pq (eiTtt)!1+1pq
1
e C.diag(t).t q e Ttt p p+q
i i
=
= (cid:18) eiCt (cid:19) (cid:18)eiCt (cid:19) !
ti.ei.Tt + 2α.ti.ei.C.diag(t) T t p t2 q (p+1q)
" (1+α)ei.Tt.t! (1+α)ei.C.diag(t).t! ti = j∈S ji j . j∈S j
" P j∈Stj ! Pj∈Stj! #
t .e .C
− i i .δtk−1 Which is equationP5 hence globaPl trust exists, and can be
ei.C.t!# calculatedbyaboveequation.
9
0.6 choosing TTL value. A requesting peer will wait for a
α=1
time greater than 2xTTL. If no response is received within
α=1/2
waiting period, the query can be made again with larger
α=1/3 TTL value. After getting the response from the network, a
0.4 α=1/4 peer can select the most reputed peer as the source peer
α=1/5 and can download the required file. In order to balance
al
u the load of the network, a peer can select the set of peers
d
si whoseglobaltrustismorethantheGlobal ref andthenthe
e
R source peer can be selected probabilistically among them.
0.2
The probability of selecting any peer as a source can be
takentobeproportionaltoitsglobaltrust.Thisstrategyhas
twofold effects, one is to allow only the reputable peer to
become the source, and another balancing the load among
0
the reputable peers. After selecting the source peer and
0 5 10 15 20 gettingthefilefromit,apeercanevaluatequalityoffileand
cansendthefeedbacktothe peersholding the trustvalues
Numberofiterations
ofsourcepeer.Hereitisimportanttonotethatiflocaltrust
ofanysourcepeerisupdatedthenitwillnoteffectthelocal
Figure 3: Convergence of Algorithm for different values of
trust of other co-source peers, and peer needs to send the
α
updatedfeedbackofonlythatsourcepeerwhoselocaltrust
is being updated. However in normalization methods [3]
5 ANALYSIS OF ALGORITHM [6], if local trust value of any source peer is updated then
local trust value of all other source peers also have to be
5.1 SpeedofConvergence
updated. Because in normalization method, sum of all the
Figure 3 shows the speedof convergence of our algorithm,
trustvaluesassignedbyanypeertoallitssourcepeershave
graph is plotted for the average value of residue of global
tobeone.Thisguaranteestheconvergenceofglobaltrustin
trustatanynodeastheiterationsperforms,i.e.
normalizationmethod.Letaveragenumberofsourcepeers
1 per peer is avg source then in one update of local trust
||tk−tk−1||
N 1 we are savingavg source−1 number of messages, which
is very significant for the whole network. In this process,
Wecanseefrom thefigurethatasαisdecreasingitsspeed
if all of the responding peers have a global trust less than
of convergence is increasing. For α ≤ 1/3, it is converging
Global ref thenallof them canbe rejectedandrequesting
inlessthenseveniterations.Lowervalueofαmeanshigher
peercango for anothersearchbyincreasingthe TTLvalue
value of p compare to q. Higher p means more weightage
ofquery.ThereshouldbeanupperlimitonTTL,afterwhich
to first term, which is weighted average of trust value
peershouldstopandterminatethequeryprocess.
informed one hop neighbors. Lower q means lesserweight
Global trust can be updated by each trust holder peer
to second term which is equivalent global trust of trust
according to algorithm 2. It is similar to update method
assigning set S. But there is trade-off between these two.
used in [3] [6]. To calculate the global trust value of any
First term is used to settle the conflicts among direct trust
peer,trust holderpeerneedstoknowthelocaltrustvalues
assigningpeers,and second term is usedtobias the global
of that peer and the current global trust value of trust
trustofpeeraccordingtoglobaltrustofthetrustassigning
assigning peers. Trust assigning peers will send the local
set. The globaltrust of individual members of set is biased
trust values of source peers to their trust holder peers and
by their trust assigning set respectively, and so on. Hence
trust holder peers will ask the current global trust values
second factor is taking the opinion from the whole of the
of trust assigning peers from their respective trust holder
network. We cannot neglect the opinionofother peersbut
peers.Thisprocessisrepeatedtilltheconvergenceofglobal
we also need faster convergence of the algorithm. Because
trust see algorithm 2. For security purpose, more than one
higher the speedof convergence, lesserwillbe the number
peers can manage the global trust of a particular peer.
ofmessageneededtoupdatetheglobaltrust.
Againhere,itisimportanttonotethatnumberofiterations
required to converge the global trust will be maximum in
5.2 ImplementationinDistributedSystem
first time only. In all successive updates global trust will
Algorithm 1 describes how the requesting peers can select converge more faster since initial guess of global trust will
thepeerfrom whomtodownload.Wewillcalltheselected be more close to the final global trust value. Global trust
peers as source peers. Each peer can set the Global ref, a willconvergeforanyinitialvalueofglobaltrustvector,t0.
reference value of global trust, to decide whether to select Butwehavetoensurethat,atleastonecomponentofglobal
a peer as a source peer or not. If global trust of any peer trust vector, t0, must be along the final global trust vector,
is less than Global ref, then it should not be selected t, see section 4. We can ensure this by taking initial global
as a source. The requesting peers initiates a query for a trustvalueforallpeersas(w g+w b)/2.
resources. Each query is given a TTL value. Whenever a So far we have discussed that how to aggregate the
query is forwarded, its TTL value is decremented. When globaltrustfromlocaltrustandinpeerselectionprocedure
TTL becomes zero, the query is not forwarded anymore. we are considering only the global trust. However global
The requesting peer can control the scope of query by trust is more significant if peer has no past history with
10
any of responding peer. If peer has some past history with Algorithm2ForupdatingtheGlobalTrustofpeers
any one of them then decision of selection of source peer 1: Input:LocalTrustvaluesofpeers
can be done according to βti + (1 − β)Tji. It is convex 2: Output:Globaltrustontrustholderpeers
combination of global trust of peer and local trust value 3: procedure
assignedbyrequestingpeertorespondingpeerinpast.The 4: for eachpeeri do
valueofparameterβ canbeselectedbypeerdependingon 5: forallpeerj,whoisselectedassourcepeerdo
itsconfidenceonrespondingpeer. 6: Evaluatethereceivedfile;
7: Assign the Local Trust value between w b to
Algorithm1Forselectionofsourcepeer w g;
1: procedure 8: Send the local Trust to trust holder peer of
2: Global ref ← (w g+w b) peerj;
2
3: TTL←Const. 9: endfor
4: top: 10: if Peeriistrustholderpeerofpeerkthen
5: SetTime Counter≥2∗TTL 11: forall peerj,whoselectedkasasourcepeer
6: i←0 do
7: SendthequeryforrequiredfileinNetwork; 12: ReceivetheLocalTrustvaluesTkj;
8: whilei≤Time Counter do 13: Locatetheirtrustholderpeer;
9: Waitforresponsefromthenetwork; 14: endfor
10: i←i+1; 15: Initialization;
11: endwhile 16: Setp,q,previous tk,threshold;
12: ifNumber of responding peers==0then 17: whileerror ≥thresholddo
13: ifTTL≥(TTL)upper then 18: ReceivetheGlobalTrusttj fromtheirtrust
14: Terminatethequeryprocess; holderpeer;
15: else 19: Compute
p q 1
1176:: gInoctroeatospeTTL; 20: tk ←" PPj∈jS∈STjtkjtj! . PPjj∈∈SStt2jj! #(p+q)
18: endif
19: else 21: error ←|tk−previous tk|
20: GettheGlobal Trustofalltherespondingpeers 22: previous tk ←tk
23: endwhile
fromtheirtrustholderpeer;
24: endif
21: SelectthepeerwithmaximumGlobal Trust;
25: endfor
22: ifGlobal Trust≥Global ref then
26: endprocedure
23: Downloadtherequiredfile;
24: Evaluatethefile;
Table2:Valuesofvariousparameterswhichweusedinour
25: Send the feedback to trust holder peer of
simulation
sourcepeer;
26: Stop;
S.N. Parameter Description Value
27: else 1 N NumberofPeersinNetwork 100
28: ifTTL≥(TTL)upper then 2 Num file NumberofdifferentfilesinNet- 1000
29: Terminatethequeryprocess; work
Total number of transaction in
30: else 3 Num transact 10000
network
31: IncreaseTTL; 4 γ Zipf’sConstant 0.4
32: gototop 5 p Seefromequation5 3
33: endif 6 q Seefromequation5 1
34: endif 7 w b Weightfactorforunsatisfactory 1
file
35: endif Weight factor for satisfactory
8 w g 10
36: endprocedure file
ThresholdvalueofGlobalTrust
9 Global ref 5.5
forgoodpeers
6 EXPERIMENTAL EVALUATION
behavioralconditionsofpeersinthenetwork.Itisexplained
Referring to [9] and [18], we used NetLogo 5.2 [19], to
innextsubsections.
evaluate the performance of our algorithm. NetLogo is a
multi-agent programmable modeling environment where
6.1 Simulationsetup
we can model the different agents and can ask them to
performthetaskinparallelandindependently.Itiswritten We simulate a typical p2p network with parameters and
mostlyinScala,withsomepartsinJava.Wealsosimulated distribution taken from real world measurements [20]. We
andcomparedourresultwithtwomostpopularreputation usedpercentageofauthenticdownloadasastandardmetric
system Eigen Trust [3] and Power Trust [6]. We found to evaluate and compare the performance of reputation
that our algorithm is giving better performance in various systems.Inthismodel,peercanissueaqueryforparticular